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Università della Svizzera italiana Scuola universitaria professionale della Svizzera italiana
IDSIA Istituto Dalle Molle di studi sull’intelligenza artificiale
Ant Colony Optimization
Theory and application to vehicle routing problems
Roberto Montemanni Dalle Molle Institute for Artificial Intelligence (IDSIA)
University of Applied Sciences of Southern Switzerland (SUPSI) Lugano, Canton Ticino, Switzerland
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Università della Svizzera italiana Scuola universitaria professionale della Svizzera italiana
IDSIA Istituto Dalle Molle di studi sull’intelligenza artificiale
Contributors:
Luca Maria Gambardella Andrea Emilio Rizzoli Alberto Vittorio Donati
Norman Casagrande
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Customers Requests Time Windows Pick-up and delivery Access Limitation
Fleet Non-homogeneous vehicles Costs (trucks own/external) Drivers Time limitation
Information Driving time Limitation on max km. Depots, number, location
Complex Vehicle Routing Problems
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Objectives: (multiples) • Total distance minimization • Travel time minimization • Number of vehicles minimization • Fleet optimization … cost function minimization
Vehicle Routing Problems
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Compute the optimal solution ?
Time Number of Operations Clients Less than 10 sec. 1'000'000'000'000 1000 mil. 40 1hour 60'000'000'000'000 6.00.E+13 46 1day 3'600'000'000'000'000 3.60.E+15 52 1 year 1'281'600'000'000'000'000 1.28.E+18 60 100 years 128'160'000'000'000'000'000 1.28.E+20 67 1000 years 1'281'600'000'000'000'000'000 1.28.E+21 70
Clients N. Solutions 2 4 4 16 8 256
16 65'536 32 4.29.E+09 64 1.84.E+19
128 3.40.E+38 256 1.16.E+77 512 1.34.E+154
1'024 1.79E+308
Evaluate all the possible combinations of customers and trucks The number of solutions grows as a function of 2n
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Time Number of Operations, 1000 time faster Clients Less than 10 sec. 1'000'000'000'000'000 1'000'000 mil. 50 1 hour 60'000'000'000'000'000 6.00.E+16 56 1 day 3'600'000'000'000'000'000 3.60.E+18 62 1 year 1'281'600'000'000'000'000'000 1.28.E+21 70 100 years 128'160'000'000'000'000'000'000 1.28.E+23 77 1000 years 1'281'600'000'000'000'000'000'000 1.28.E+24 80
Compute the optimal solution ? Clients N. Solutions
2 4 4 16 8 256
16 65'536 32 4.29.E+09 64 1.84.E+19
128 3.40.E+38 256 1.16.E+77 512 1.34.E+154
1'024 1.79E+308
Evaluate all the possible combinations of customers and trucks The number of solutions grows as a function of 2n
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• The main problem is how to manage complexity.
• The idea is to optimize in a few seconds using adaptive search algorithms which dynamically learn new strategies
How can we efficiently solve them?
A recent approach is to combine ideas coming from biology, computer science,
operational research and artificial intelligence [Metaheuristics]
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First they explore Individual ants mark their path by emitting a chemical substance - a pheromone - as they forage for food
Ants smell pheromone and they tend to choose path with strong pheromone concentration Other ants use the pheromone to find the food source When the “system” is interrupted, the ants are able
to adapt by rapidly adopting second best solutions Social insects, following simple, individual rules, accomplish complex colony activities through: flexibility, robustness and self-organization
How ants search for food
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Ants compute the shortest path
Two bridges experiment: from nest to food constrained to move in two asymmetric paths (Goss, 1989)
Ants (termites) can smell Pheromone trails
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colony
food
nest
pheromone
fleet
goods
depot
Optimized Solution
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• ACO algorithms are multi-agent systems that exploit artificial stigmergy for the solution of combinatorial optimization problems.
• Artificial ants live in a discrete world. They construct solutions making stochastic transitions from state to state.
• They deposit artificial pheromone to modify some aspects of their environment (search space). Pheromone is used to dynamically store past history of the colony.
• Artificial Ants are sometime “augmented” with extra capabilities like local optimization
ACO algorithms for VRPs
Artificial Ants
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Given an optimization problem
Lugano�
Mendrisio�
Cost km
Artificial Pheromone
Artificial Ants build solutions using cost and pheromone
Best solutions are rewarded with new pheromone
The Algorithm learns from experience to compute always better solutions
ACO algorithms for VRPs
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State transition rule
s =
arg maxu∈Jk r( )
τ r,u( )[ ]⋅ η r,u( )[ ]β{ } if q≤ q0 (Exploitation)
S otherwise (Exploration)
'
( )
* )
Where: • S is a stochastic variable distributed as follows: • τ is the pheromone • η is the inverse of the distance • Jk(r) is the set of cities still to be visited by ant k positioned on city r • β and q0 are parameters
pk(r,s) =
τ (r,s)[ ]⋅ η(r,s)[ ]β
τ (r,u)[ ]⋅ η(r,u)[ ]β
u∈Jk (r)∑
if s∈Jk(r)
0 otherwise
'
( )
* )
ACO algorithms for VRPs
.
?5
15
80%+1520
⋅ 20%= 95%
520
⋅ 20%= 5%
q0=0.8
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Pheromone updating
( ) ( ) ( ) 0,1, τρτρτ ⋅+⋅−= srsr
τ r,s( )← 1−α( ) ⋅τ r, s( ) +α ⋅Δτ r,s( )Global
Global updating: at the end of each cycle
Local Updating: when edge (r,s) is visited by an ant
where Δτ r,s( )Global = 1Lbest
ACO algorithms for VRPs
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Since 1994 12 Swiss FNS projects
5 European projects 4 Swiss CTI project
Università della Svizzera italiana Scuola universitaria professionale della Svizzera italiana
IDSIA Istituto Dalle Molle di studi sull’intelligenza artificiale
First competitive results extending Ant Colony Optimization with AI and OR (Gambardella and Dorigo, 1994)
Best known algorithms on TSP, VRP, QAP, SOP, TOP (these are routing and scheduling problems)
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The Ant Colony algorithm
• Loop • position all ants in depot • For step=1 to n • For ant=1 to m • Find a “feasible” order • Select the next order by using exploration and exploitation • Apply the local trail updating rule • Apply local search • Apply the global trail updating rule • Until End_condition
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Ant Colony Optimization diffusion
New ACO algorithms
International reviewers check for validity
Comparison with benchmarks
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First prototype in a realistic scenario Customers ask for fuel delivery at home (house heating) Accessibility restrictions
Dynamic fleet optimization for fuel distribution
Pina Petroli, Grancia
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Fleet Non-homogeneous trucks (different size and configuration) Half day availability Half day planning with one week visibility Data exchange between the main station and the trucks
Dynamic fleet optimization for fuel distribution
Pina Petroli, Grancia
Operating area: Canton Ticino, Switzerland
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The optimisation modules
• OFF line planning: – finds the best distribution routes for the vehicle fleet
on the basis of the previous day‘s order • ON line planning:
– re-routes the fleet in function of urgent delivery requests received during the distribution process
Dynamic fleet optimization for fuel distribution
Pina Petroli, Grancia
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A tour in the Sopraceneri area (Canton Ticino, Switzerland)
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Improvements certified by
Pina Petroli SA, Grancia, CH
Computational time PC: 3 minutes Average improvement: 20%
Dynamic fleet optimization for fuel distribution
Pina Petroli, Grancia
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IDSIA
Der Spiegel
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Distribution problem
• Central Depot • Non-homogeneous fleet • Customers accessibility
restriction • Customers time windows • Operations area:
Switzerland
Objectives
1. N. of tours minimization 2. Cost minimization
Cost function = total_km * km_cost + total_time_violation * tv_cost
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Non-homogeneous fleet of vehicles
• Truck
• Truck + trailer
• Tractor unit + semi-trailer
Other features:
• Embedded lift
• Refrigerated container
• Parking time (constant)
• Service time (variable)
• Trailer hook / unhook time (constant)
Remark: All capacities are expressed in pallets.
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Ant-Route as a Strategic tool Total number of tours
50
55
60
65
70
75
80
85
90
10 30 60 90 120 180 240 infinite
Time windows width (min)
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MOSCA EU
On line urban distribution with dynamic traffic
information
IT, DE, CH, UK
LAN
Ethernet
Eth
erne
t / O
ptoc
al F
iber
Terminal Control Centre
Terminal Control Centre
Terminal Management
CommunicationNetwork
Gate Control
Gate system
Other Gates
Booking & Reservation
Yard Control Forwarding Companies
Radio tower
CommunicationNetwork
TranshipmentEquipment
Outside theterminal
Inside theterminal
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Planning Urban Activities
8.00 am
2.00 pm
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Time Dependent VRP
Given a VRP Problem
To better model traffic condition (this is crucial in urban distribution)
costs on the arcs are time dependent
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TD-VRP Search Space
TIME
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TIME Departing time
spee
d
10.00 8.00 12.00
2
1
Departing time
Trav
el ti
me
10.00 8.00 12.00
1
.5
TD-VRP Search Space
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Planning under different traffic conditions
• Freight distribution in urban environment • Testbed: the city of Padua, Nothern Italy • Technically there is a different Ant Colony for
each time interval.
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Padua, Italy
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Planning under different traffic conditions
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Planning under different traffic conditions
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The Dynamic Vehicle Routing Problem (DVRP)
• New orders arrive when the working day has already started
• New orders have to be assigned to vehicles which may have already left the depot
• A communication system must exist between vehicles and the depot
• Vehicles do not need to go back to the depot when they are assigned new orders
Problems covered:
• Parcel collection • Feeder systems
• Fuel distribution • ...
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Strategy for DVPRs
Working day
• The working day is divided into nts time slices
• For each time slice a Static Vehicle Routing like Problem is solved by an Ant Colony Optimization (ACO) algorithm
0 T
Time slice
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The ACS-DVRP algorithm. Elements
3. Pheromone conservation procedure • Passes information about good solutions from a SVRP to the following one of the sequence
1. Event manager • Collects new orders • Keeps trace of the already served orders • Keeps trace of the current position of vehicles. • Creates a sequence of SVRPs • Assigns orders to vehicles
2. Ant Colony System (ACS) algorithm • Solves SVRPs
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ACS-DVRP - Overview
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ACS-DVRP - Event Manager (1)
Main element of the architecture. It co-ordinates the other modules accordingly to these rules: • The working day is divided into nts time slices, each one long Tns = T / nts seconds • Orders received after the time of cutoff Tco are postponed to the following day • Orders received during a time slice are processed only at the end of the time slice
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ACS-DVRP - Event Manager (2)
At the end of a time slice the following operations are carried out: • Orders starting during the next time slice (in the solution of the last SVPR) are committed to the respective vehicles
• A new SVR-like problem is created, where – New starting positions and residual capacities are calculated for the vehicles – New orders received during the last time slice are inserted and committed orders are deleted
• The pheromone conservation strategy is run
• The ACS algorithm is run until the end of the next time slice
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ACS-DVRP - Pheromone conservation
When the ACS algorithm finishes working on a SVRP: • Pheromone matrix contains encrypted information about good solutions • The next SVRP of the sequence is potentially very similar to the SVRP just considered
These considerations are used to prevent optimization to restart each time from scratch. The new pheromone matrix is then as follows: τij = (1- γr) τijold + γrτ0 for each pair of customers contained
in both the new and the old SVRP τij = τ0 for pairs involving new customers γr is a new parameter which regulates pheromone conservation
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Computational results
Min Max Avg Min Max Avg
c100 1080,33 1169,67 1124,04 973,26 1100,61 1066,16
c100b 978,39 1173,01 1040,99 944,23 1123,52 1023,60
c120 1546,50 1875,35 1752,31 1416,45 1622,12 1525,15
c150 1468,36 1541,54 1493,06 1345,73 1522,45 1455,50
c199 1774,33 1956,76 1898,20 1771,04 1998,87 1844,82
c50 693,82 756,89 722,15 631,30 756,17 681,86
c75 1066,59 1142,32 1098,85 1009,38 1086,65 1042,39
f134 16072,97 17325,73 16866,79 15135,51 17305,69 16083,56
f71 369,26 437,15 390,48 311,18 420,14 348,69
tai100a 2427,07 2583,02 2510,29 2375,92 2575,70 2428,38
tai100b 2302,95 2532,57 2406,91 2283,97 2455,55 2347,90
tai100c 1599,19 1800.85 1704,40 1562,30 1804,20 1655,91
tai100d 2026,82 2165,39 2109,54 2008,13 2141,67 2060,72
tai150a 3787,53 4165,42 3982,24 3644,78 4214,00 3840,18
tai150b 3313,03 3655,63 3485,79 3166,88 3451,69 3327,47
tai150c 3090,47 3635,17 3253,08 2811,48 3226,73 3016,14
tai150d 3159,21 3541,27 3323,57 3058,87 3382,73 3203,75
tai75a 1911,48 2140,57 2012,13 1843,08 2043,82 1945,20
tai75b 1634,83 1934,35 1782,46 1535,43 1923,64 1704,06
tai75c 1606,20 1886,24 1695,50 1574,98 1842,42 1653,58
tai75d 1545,21 1641,91 1588,73 1472,35 1647,15 1529,00
Total 53454,54 59060,81 56241,50 50855,94 57645,52 53784,02
ProblemNo pheromone ACS-DVRP
• No pheromone = multi- start local search algorithm • ACS-DVRP = the method we propose
Travel times. 5 runs for each problem (Intel P4 1.5 GHz)
ACS leads to the following improvements: • 4.86% for Min • 2.40% for Max • 4.37% for Avg
ACS has always the best values for Min and Avg
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Lugano, Switzerland
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A case study – the city of Lugano, Switzerland
Road network of Lugano with the depot (white square) and the 50 customers (black circles)
• Working day (= computation time): 8 hours • Ordered quantities in [1, 31]
• Service time: 10 minutes (for each customer) • 10 vehicles with capacity 160
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A case study - Number of time slices nts
Experiments with different values for the number of time slices nts have been carried out with algorithm ACS-DVRP
• nts = 25 is the best choice for the number of time slices
• ACS-DVRP has been successfully used in a realistic case study
11201 5760 5 10733 2880 10 9744 1152 25 10399 576 50 12422 288 100 12702 144 200
Total travel time (sec)
Time for each time-slice (sec)
Number of time-slices
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Number1 is the largest Italian logistic operator (Barilla group) Moves goods from factories to stores 1000/1300 vehicles x day No own fleet but all external trucks Multiple starting points Pick-up and delivery along Italy
Number1 Logistics Group Italia
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Performances Planner VS AntRoute
78.00%
80.00%
82.00%
84.00%
86.00%
88.00%
90.00%
92.00%
94.00%
Giorno
1
Giorno
2
Giorno
3
Giorno
4
Giorno
5
Giorno
6
Giorno
7
Giorno
8
Giorno
9
Giorno
10
Giorno
11
Giorno
12
Giorno
13
Giorno
14
Giorno
15
Giorno
16
Giorno
17
Periodo di test
Ren
dim
ento
RendimentoAntRoute
Rendimentomanuale
+4%
+2%
+6%
+8%
+10%
-6%
-4%
-2%
Manuale
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Other VRP Applications
HUGO BOSS: Green Supply Chain Optimization
CARITAS CH: Exhausted oil collection, recycled then to Bio-fuel
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Basic Research On-line Fleet Management
Extension of the ACO approach to Stochastic and Robust TSP and VRP problems
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Thank for your attention!
Università della Svizzera italiana Scuola universitaria professionale della Svizzera italiana
IDSIA Istituto Dalle Molle di studi sull’intelligenza artificiale
www.idsia.ch - [email protected]